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Although sometimes it seems like you can- if there is not already a revealed tile on that space. I am understanding this correctly?

The idea is that tiles that move into the "side position" or decks, are moved below the the others. Meaning a tile with a character on can't be moved below another tile. The other case is when there is no tile in the side position on the other side of a row/column. Then a tile can't be moved further to the opposite side.